Many state DOTs hire third party inspectors to watch over their projects. Many state DOTs require the testing agency to use CWIs to act on their behalf as third party inspectors. Remember, if that is the situation in your case, the CWI may have been one of those individuals that passed the examinations by one or two points.
Remember, the fellow that graduates at the bottom of his medical class is still called "Doctor". You wouldn't want that individual to operate on you, but he/she is still a doctor.
A CWI that passes the examination by scoring at least 72 on the examinations is still called a CWI. He or she are qualified to act as the state's TPI on bridge projects. Scary, but true.
Brent is correct in saying it is better to work with the TPI when you can, but there are those cases where one must say, "Sorry buddy, but you are wrong." The TPI should be able to show you where you turned left when you turned right. If he says "It is the way I want it", then you have a case. Do it in a nice way, you don't need another enemy.
Best regards - Al